Tricky Trigonometry

Geometry Level pending

\(x\) is a real number. Let \(S\) be the sum of all the possible distinct values of \(\sin x\) that satisfy the following equation:

\[7 - 2\cos^2x - 11\sin x = 0\]

\(S\) can be written as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

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